Finite Group Databasebeta

← #11737867

Subgroups — #11737867

Order 896.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90412C211yesmaximal, characteristic, complemented
2#90233C211yesmaximal, characteristic
3C7×#461C211yesmaximal, characteristic
4#89939C212yesmaximal
5#89938C212yesmaximal
6#89864C211yesmaximal, characteristic, complemented
7C4×#1142C211yesmaximal, characteristic
8#89621C211yesmaximal, characteristic, complemented
9#89603C211yesmaximal, characteristic, complemented
10#89565C212yesmaximal, complemented
11#89561C212yesmaximal, complemented
12C2×C7×SD16C2212yes—
13C7×C4⋊Q8C2211yescharacteristic
14C4×C7×D4C2211yescharacteristic
15Q8×Dic7C2212yes—
16#6507C2212yes—
17C2×#1144C2212yes—
18C2×#1144C2212yes—
19#6499C2212yes—
20C2×#1142C2211yescharacteristic
21#6489—21no—
22#6487—21no—
23C2×Dic28C2212yes—
24#6453—21no—
25#6452—22no—
26#6450—21no—
27#6450C2212yes—
28#6442—22no—
29#6432C2211yescharacteristic
30C4×Dic14C2211yescharacteristic
31C4×Dic14C2211yescharacteristic
32C7×C4⋊C8C2211yescharacteristic
33C7×#103C2212yes—
34#6406C2211yescharacteristic
35#6390C2212yes—
36#6389C2212yes—
37Dic7⋊C8C2212yes—
38#6384C2211yescharacteristic
39#6382C2211yescharacteristic
40C4×C7⋊C8C2211yescharacteristic
41#3398—71nomaximal, Sylow
42C2×C7×Q8C2312yes—
43C2×C7×D4C2311yescharacteristic
44C22×C4×C7—21no—
45C2×Dic14D411yescharacteristic
46C2×Dic14C2311yescharacteristic
47C2×Dic14C2312yes—
48C7×SD16—42no—
49C2×C7×C8C2312yes—
50C7×C4⋊C4—21no—
51C7×C4⋊C4—21no—
52C7×C4⋊C4C2311yescharacteristic
53C7×C22⋊C4—21no—
54C42×C7C2311yescharacteristic
55C22⋊Dic7—22no—
56C22⋊Dic7—22no—
57#1144—24no—
58#1144—42no—
59#1142—22no—
60C4⋊Dic7—21no—
61C4⋊Dic7—21no—
62C4⋊Dic7—22no—
63C4⋊Dic7D411yescharacteristic
64C4⋊Dic7C2311yescharacteristic
65Dic7⋊C4—21no—
66Dic7⋊C4—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4D412yes—
70C4×Dic7—21no—
71C4×Dic7—22no—
72C4×Dic7—22no—
73C4×Dic7C2312yes—
74C2×C7⋊C8C2311yescharacteristic
75C2×C7⋊C8C2311yescharacteristic
76Dic28—42no—
77#557—71no—
78#556—71no—
79#497—72no—
80#494—71no—
81#464—72no—
82#462—71no—
83#462—72no—
84#461D711yescharacteristic
85#446—71no—
86#441—72no—
87C4×SD16—71no—
88C23×C7—21no—
89C7×Q8—22no—
90C7×Q8—22no—
91C7×D4—21no—
92C7×D4—21no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—42no—
96C2×C4×C7C2×D411yescharacteristic
97C2×C4×C7C2×D411yescharacteristic
98C2×C4×C7C2411yescharacteristic
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7C2×D412yes—
103C2×Dic7C2×D412yes—
104Dic14—22no—
105Dic14—21no—
106Dic14—22no—
107Dic14—21no—
108Dic14—22no—
109C7×C8—22no—
110C7⋊C8—21no—
111C7⋊C8#4112yes—
112C2×Q16—72no—
113C2×Q16—72no—
114C2×Q16—72no—
115C2×SD16—71no—
116C2×SD16C2×D712yes—
117C4⋊Q8—72no—
118C4⋊Q8—71no—
119C4⋊Q8C2×D711yescharacteristic
120#126—142no—
121#125—142no—
122#125—141no—
123#124—72no—
124C22⋊Q8—141no—
125C4×Q8—71no—
126C4×Q8—72no—
127C4×Q8—72no—
128C4×Q8—71no—
129C4×Q8—141no—
130C4×D4C2×D711yescharacteristic
131#117—141no—
132#106—71no—
133C4⋊C8—72no—
134C4⋊C8C2×D711yescharacteristic
135#103—72no—
136#103—71no—
137#103C2×D712yes—
138#102—71no—
139#97—72no—
140C4×C8—71no—
141C22×C7—41no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7C22×D411yescharacteristic
145C4×C7—21no—
146C4×C7—42no—
147C4×C7—41no—
148C4×C7#13712yes—
149C4×C7C2×#4111yescharacteristic
150C4×C7H(5,𝔽2)11yescharacteristic
151Dic7—22no—
152Dic7—22no—
153Dic7—41no—
154Dic7—42no—
155Dic7—41no—
156C2×Q8—71no—
157C2×Q8—72no—
158C2×Q8—71no—
159C2×Q8C22×D712yes—
160C2×D4C22×D711yescharacteristic
161C22×C4—21no—
162Q16—144no—
163Q16—282no—
164Q16—282no—
165SD16—42no—
166SD16—142no—
167C2×C8—71no—
168C2×C8—71no—
169C2×C8C22×D712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—72no—
173C4⋊C4—71no—
174C4⋊C4—71no—
175C4⋊C4—142no—
176C4⋊C4—142no—
177C4⋊C4—142no—
178C4⋊C4—142no—
179C4⋊C4—141no—
180C4⋊C4—141no—
181C4⋊C4—141no—
182C4⋊C4C22×D711yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—142no—
185C22⋊C4—142no—
186C42—72no—
187C42—142no—
188C42—142no—
189C42—141no—
190C42C22×D711yescharacteristic
191C2×C7—81no—
192C2×C7#55011yescharacteristic
193C2×C7C2×#13711yescharacteristic
194C2×C7#57811yescharacteristic
195C23—21no—
196Q8—22no—
197Q8—22no—
198Q8—142no—
199Q8—141no—
200Q8—142no—
201Q8—141no—
202Q8—142no—
203D4—21no—
204D4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—72no—
209C2×C4—72no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4D4×D711yescharacteristic
214C2×C4D4×D711yescharacteristic
215C2×C4C23×D711yescharacteristic
216C8—22no—
217C8—72no—
218C8—141no—
219C7#339811yescharacteristic, complemented, Sylow
220C22—41no—
221C22—41no—
222C22—41no—
223C22C2×D4×D711yescharacteristic
224C4—21no—
225C4—41no—
226C4—42no—
227C4—142no—
228C4—142no—
229C4—281no—
230C4—282no—
231C4—281no—
232C4#647812yes—
233C4#654811yescharacteristic
234C4D7×#4111yescharacteristic
235C2—81no—
236C2#9029011yescharacteristic
237C2C2×#647811yescharacteristic
238C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#461C2
socleC22×C7C22×D4
prime radicalC2×C7×D4C23
hypercenter#461D7
hypocenterC7#3398
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #90412C2semidirect
6. #89864C2semidirect
8. #89621C2semidirect
9. #89603C2semidirect
10. #89565C2semidirect
11. #89561C2semidirect
219. C7#3398semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#904122
#902332
C7×#4612
#899392
#899382
#898642
C4×#11422
#896212
#896032
#895652
#895612
#33987